Best you could do is suggest that the player turns them on through a tip or text message. My guess is that Blizzard won't allow us to access the players main settings, which is where the subtitle option is stored. This is for security reasons, as a random map could intentionally change the players settings, messing them all up. It's the same reason we are not able to pull the players name as a string, only as text.
Edit: Alternatively, you could create a dialog with the background hidden and use that to show the subtitles. It's more work, but it would make sure that everyone playing the map views your subtitles without issue. It also gives you full creative control, as you could design it however you want without having to deal with the XML UI files.
I don't know exactly what is required for the subtitles to appear, but one thing I figured out is that they automatically deactivate on battle.net when a sound is played during a transmission (when subtitles are turned off in the options).